Let's start with the actual keyboard, itself.
Digital pianos, unlike most electronic keyboards, have weighted keys, designed to mimic the feel and action of an acoustic piano.
This is sometimes described in the manufacturer's brochure, as a 'Weighted, graded, hammer action'
In other words, the keys are weighted, and the amount of pressure needed to strike the keys, is graduated from the lower notes, to the highest notes, with the lower keys requiring more pressure to play them, than the higher ones, just like on an acoustic piano.
'Hammer action', means that the mechanical parts, inside, also act in a similar way to those of an acoustic piano, rather than using springs, as in most electronic keyboards.
As the price increases, so generally, does the quality of the keyboard action, with some of the higher end models, having solid wood keys, and even a simulated ivory facing , which attempts to copy the feel and friction of real ivory keys.
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The Piano Sound

The piano sound is created, by recording samples of a high quality acoustic piano, and storing those samples in a memory chip inside the piano, ready to be replayed when the keys are struck.
This is a very simplistic description of how the sound is created, as recording the original samples, is a highly specialised field, but generally, the larger the number of samples used ( these include recording samples of keys struck at varying velocities, the sound of the key being released, etc) the more faithful the digital piano will sound.

It is almost impossible to completely reproduce a true acoustic piano sound, but digital pianos are getting better all the time.
For example, when a key is played on an acoustic piano, the hammer hits the string, and this also causes nearby strings to resonate, contributing to the acoustic piano's unique sound. Also, the soundboard, and the case of the piano resonate, and this is extremely difficult to reproduce electronically.
Some digitals now have a soundboard built in, with special microphones to pick up this resonance, and replay it through the piano's sound system, to get as close to the real thing as possible.

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Polyphony

Polyphony, simply means the number of notes that can be played at the same time.
When playing some piano pieces, many notes may be played, very fast, and also, using the sustain pedal at the same time, to hold some of those notes, can quickly use up much of the polyphony available.
When this polyphony limit is exceeded, certain notes will be cut off, to allow more notes to be played, so, the higher the polyphony number, the better.

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MIDI

MIDI stands for Musical Instrument Digital Interface.
So what is it, and why would I need it?

The MIDI connectors on a digital piano, provide a way to connect the instrument to a computer, or to an external sound module.
You would connect your piano to a computer, if you wanted to use midi editing, or sequencing ( for sequencing, read recording) software, to record and manipulate your performances on a computer.

An external sound module is simply a box of electronics, which contains more sampled intrument sounds, which may be different to, or of a higher quality than, those built into your piano.

MIDI does not transmit audio, or sound, but only electronic data, such as which key was pressed, how hard, and for how long, and what instrument sound is to be used.
This data can be sent to a computer containing it's own sound module, or to an external sound module, and the data will play those sounds.

Most owners, unless they are seriously interested in comuter editing and recording, will probably never use the external midi features of their instrument, although it is a fairly standard feature on most digital pianos, it is not essential.

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Sequencers and Recording

As I explained earlier, a sequencer, is built in software, that can record your performance.
Basically, it records the MIDI data of your performance, stores it in the piano's memory, and it can be played back at the touch of a button, exactly as you played it.
These can be useful while learning, as you can record your left hand part, for example, and play it back, while you practice the right hand part.
Multi-track sequencers, can record generally up to 16 performances, so you could for example, record your piano performance, and while it is playing back, you could record a string part on another track, gradually building up the piece to a multi instrument performance.
Some pianos provide sequencers, with extensive editing features, enabling you to mix the volume levels of each track, and even correct a wrong note!

If you are serious about multi-track recording and editing of your music, this is where the MIDI feature is useful, as you can edit your music on a computer screen, much more easily than on the tiny display built into the piano.
Some pianos have floppy disk drives, so that you can transfer your MIDI performance to the computer more easily, but remember, only MIDI data will be recorded, and not audio sounds. You will need to have those sounds already installed in your computer.
The floppy disk drive can also be used to transfer MIDI files from your computer (many of which can be freely found on the internet) to your piano, where they will play back the MIDI file, using the piano's internal sounds, turning your instrument into a home entertainment centre,

On the higher end pianos, you may find, as well as a MIDI sequencer, an audio recorder, that will allow you to record the actual sounds of the piano (the audio) onto a built- in CD recorder, or the piano may have a connection for a USB thumb drive, to transfer the audio performance to your computer, so that you could convert the files into CDs, or audio MP3s, to post on the internet, or send to family and friends.

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Pedals

Normally, you will find either two, or three pedals on a digital piano.

In the case of two pedals, the one on the right, will be the 'sustain' pedal.
When this is depressed, it will cause the notes to keep on sounding, before gradually dying away, allowing more expression to be added to your perfomance.
The left pedal has the opposite effect, and softens, or damps the notes, so that they sound more muted.

On a three pedal piano, the centre pedal is known as the 'Sostenuto' pedal.
This only sustains the sound of selected keys being pressed, as the pedal is depressed. When the keys are released, and the pedal is held down, those notes will keep on sounding, until you take your foot off the pedal. The sustain pedal can be used while the centre pedal is held down without altering the originally held notes.
This all takes a lot of practice to perfect, but is a very useful feature, as your playing becomes more advanced.

On digital pianos, these three pedals can also be set to carry out various other functions, depending on the make and model, such as volume control, pitch bend (where a note can be made to glide up or down smoothly to the next note), adding effects, such as breath sounds, or key clicks, when playing brass or woodwind instruments, string plucking sounds, and many others.
The pedals may also be used to start and stop recording. Again, each manufacture has different alternative uses for these pedals.

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There are as many different features on digital pianos, as there are makes and models, such as extra instrument sounds (you may see the terms 'GM sounds', or 'XG sounds'. These are simply extra MIDI instrument sound groups, that all follow a standard MIDI specifaction, so that any MIDI files played back through the piano, will all use the correct instrument sounds. These sounds can also be played individually on the piano, and used with the sequencer, to record multi-track performances.
Some pianos have hundreds of extra instrument sounds, drum kits, pre-programmed backing tracks and rhythms. Metronomes, built in teaching aids, chords which can be played using a single finger, karaoke features, vocal recording, effects,..... the list goes on.
In the end, it all comes down to buying a piano that gives you what you need or want in an instrument.
Sometimes, including all the bells and whistles, and at the same time, trying to keep the price within a certain range, means that manufacturers have to compromise on the quality of the basic piano sound.
Just keep in mind what you want out of your digital piano, don't be pressured into paying extra for features, you may never need, or use.